I'm looking for a formula can be used to calculate the magnitude of a passing satellite (e.g. the ISS) based on it's TLE

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    $\begingroup$ There isn't any formula like that. Apparent magnitude depends on reflectivity index and atmospheric conditions, with former also depending on satellite's current configuration such as solar panels orientation w.r.t. the observer. None of that is given in the two-line element set (TLE). You could establish beta angle and estimate how much of incident light reflects towards the observer, but you can't make that a deterministic process. Heuristic approach might work pretty well, but don't forget to link that with observers' current observational conditions, such as weather, azimuth,... $\endgroup$
    – TildalWave
    Aug 10 '14 at 0:09
  • $\begingroup$ heavens-above.com/faq.aspx $\endgroup$
    – PearsonArtPhoto
    Aug 10 '14 at 10:14
  • $\begingroup$ You could set up a small telescope and perform photometry with a CCD. Also, it would be difficult to estimate the magnitude of the ISS with a larger aperture etc. because it's extended and not quite uniform in shape. Many amateurs have taken up imaging the ISS with their planetary cameras, such as Emil here: astrokraai.nl/viewimages.php?category=9 $\endgroup$
    – bjd2385
    Aug 10 '14 at 23:24
  • $\begingroup$ There are formulae for this, Heavens Above has one (but doesn't publish it) as well as other apps and websites. The simplest one I've been able to find for the ISS is: mag = -1.3 + 5 * log(range/1000) where range is the distance from the observer to the object in kilometers. The constant of -1.3 is based on the observed magnitude of the ISS at 1000km. The constant of 5 takes into account the size and reflectivity of the station. Is a perfect formula available, of course not. Weather is always a var, as is the amount of sunlight still in the sky. Satellites also are not perfect spheres. $\endgroup$
    – rtphokie
    Aug 12 '14 at 17:41
  • $\begingroup$ @rtphokie Those assume static reflectivity, and that simply isn't the case for satellites in LEO. There are numerous reasons why they would rotate solar panels differently even between two consecutive orbits. I have loads of long exposure ISS passes on photos that attest there doesn't seem to be any predictable pattern to it. So unless you can get also data on its current configuration, TLE alone won't help so much. Unless you're fine with large margin of error, of course. And ISS even uses two-sided panels, so they'd actually have less reasons to rotate them, but they still do. $\endgroup$
    – TildalWave
    Aug 12 '14 at 20:37

In general, this is a very hard problem. You might take a look at this paper, which looks at heuristic models for estimating satellite visual magnitude as a function of phase angle, range, object size and object type. Though IIRC, it's only looking at geosynchronous objects.

  • $\begingroup$ As I said, the basic formula above is not perfect and perfection is not really achievable. $\endgroup$
    – rtphokie
    Aug 13 '14 at 21:18
  • $\begingroup$ There are levels of accuracy that can be achieved, depending on resources. In a perfect world, you would have a 3D satellite model complete with materials & albedo, knowledge of its orientation & configuration at the time of observation, etc. With that, and lots of computing power you could come up with a very good estimate of the visual magnitude. Very few people have that luxury. But with some basic estimates of object size & albedo, there's been some work done with diffuse sphere and specular models that may at least get you in the ball park. $\endgroup$ Aug 14 '14 at 18:09
  • $\begingroup$ Not sure why the question got down-voted. It's an interesting question, and the subject of ongoing research. $\endgroup$ Aug 14 '14 at 18:13

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